Physics-Informed Diffusion Learning

• PILD is a framework that infuses generative diffusion models with physical laws like PDEs and ODEs to produce admissible and physics-consistent fields.
• It leverages innovations such as conditional embedding modules and multi-scale attention to integrate boundary, geometric, and residual constraints directly into the model.
• The approach enables rapid physics discovery, efficient simulation, and robust performance across diverse scientific and engineering applications.

Physics-Informed Learning via Diffusion (PILD) is an emergent framework that unifies generative diffusion modeling with enforcement or discovery of physical laws—such as partial differential equations (PDEs), ordinary differential equations (ODEs), and other physical constraints—within the model architecture, training objective, and inference process. Instead of treating scientific modeling and simulation as strictly equation-driven or data-driven tasks, PILD reconceptualizes the problem as conditional generative modeling, directly sampling or reconstructing physically admissible solutions under observational, geometric, or boundary constraints. PILD is instantiated in several forms, encompassing both purely data-driven physical law discovery (Camburn, 13 Jul 2025) and explicit, first-principles-constrained generative modeling (Zeng et al., 29 Jan 2026), and operating at the level of solution fields, learned operators, model parameters, and even neural network weights (Cheng et al., 31 May 2025).

1. Formulation and Theoretical Foundations

PILD fundamentally synthesizes two previously disjoint paradigms: score-based/diffusion generative modeling and physics-informed learning. The core generative model is the denoising diffusion probabilistic model (DDPM), where a physical sample $x_0$ is corrupted by progressive Gaussian noise through a forward Markov chain: $$q(x_{1:T}|x_0) = \prod_{t=1}^T \mathcal{N}(x_t; \sqrt{1-\beta_t}\, x_{t-1}, \beta_t I)$$ with corresponding reverse (denoising) process parameterized as

$$p_\theta(x_{t-1}|x_t, \mathcal{O}) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t, \mathcal{O}), \Sigma_t)$$

where $\mathcal{O}$ may encode physics information, boundary data, or other conditions.

Physics is integrated by defining a physical residual operator $\mathcal{R}(x)$—e.g., the divergence or the PDE residual at each grid point—and enforcing its minimization (or exact vanishing) through a probabilistic virtual residual observation. For Laplacian residuals, the likelihood becomes $\mathrm{Laplace}(r; \mathcal{R}(x), b_t I)$, yielding a training loss

$$\mathcal{L}_{\mathrm{PILD}}(\theta) = \mathbb{E}_{t,x_0,\epsilon}\left[ A(t) \|\epsilon - \epsilon_\theta(x_t, t, \mathcal{O})\|^2 + G(t) \left\|\mathcal{R}(x)\right\|_1 \right]$$

where $x$ is a (DDIM or denoised) sample at step $t$, $A(t)$ and $G(t)$ are schedule weights (Zeng et al., 29 Jan 2026).

This yields a modular objective that accommodates ODE/PDE residuals, algebraic constraints, and inequality penalization via the ReLU composition.

2. Architectural and Algorithmic Innovations

State-of-the-art PILD architectures employ both convolutional and transformer-based backbones, augmented by two principal innovations for physical consistency and universality (Camburn, 13 Jul 2025):

• Conditional Embedding Modules (U-FiLM/U-Att): Physical conditions $\mathcal{O}$ are injected as deep embeddings into every denoising block via feature-wise linear modulation or cross-attention, ensuring persistent enforcement of boundary or input constraints.
• Spatial Relationship and Multi-Scale Attention: Physical locality, geometric edge structure, and neighborhood interactions are encoded via pairwise positional embeddings and multi-scale neighborhood attention, biasing self- and cross-attention mechanisms to respect physical interaction ranges and symmetries.

Advanced architectures exploit VAEs for latent compression, cross-attentional boundary injection at all spatial resolutions, and Layer-wise Relevance Propagation (LRP) for analysis of emergent physics within learned representations.

3. Physical Constraints, Residuals, and Loss Design

Physics is integrated at multiple levels in different PILD formulations:

• Direct Residual Penalty: A residual operator $\mathcal{R}(x)$ is explicitly minimized along with the denoising objectives, using squared ($\ell_2$) or absolute ($\ell_1$) loss, Gaussian or Laplace virtual observation, and physically informed schedules (Bastek et al., 2024, Zeng et al., 29 Jan 2026).
• Conditional Physics Guidance: For instance, in conditional generative tasks, spatial boundary sketches, edge maps, or low-fidelity fields are used as conditioning tokens, enforcing geometric or boundary constraints throughout reverse sampling (Camburn, 13 Jul 2025, Li et al., 6 Apr 2025).
• Guided Sampling/Inference: Physics-based gradients (e.g., PDE residuals, energy gradients) are used to guide reverse diffusion trajectories during inference, ensuring samples remain close to the underlying physical manifold (Shu et al., 2022, Qiu et al., 2024).
• Parameter-Space Diffusion: Physics is enforced at the level of neural network parameters in approaches such as DiffPINN, where a diffusion model generates PINN weights aligned with physical regimes (e.g., velocity models), dramatically accelerating convergence (Cheng et al., 31 May 2025).

In multi-stage or composite models such as RMDM (Jia et al., 31 Jan 2025), physics is first imposed via a PINN-guided network or PDE-constrained U-Net, and then refined by a diffusion denoiser with anchored attention to the physics-consistent initial estimate.

4. Practical Implementations and Evaluation

PILD has been instantiated across a range of scientific and engineering problems, including:

• Universal Steady-State PDE Solvers: Direct BC-to-field mapping for electromagnetics, fluids, and mechanics using sketch-conditioned diffusion transformers and cross-attention (Camburn, 13 Jul 2025).
• Seismic and Wavefield Modeling: Diffusing over PINN parameter space to accelerate inversion and modeling workflows for variable velocity/regime (Cheng et al., 31 May 2025).
• Radio Map Reconstruction: Dual U-Net pipeline with PINN alignment and physics-guided denoising for RF mapping under sparse observations (Jia et al., 31 Jan 2025).
• Computational Fluid Dynamics Acceleration: Physics-guided reconstruction and super-resolution for Navier–Stokes, leveraging multi-scale losses and inverse-SNR scheduling (Li et al., 6 Apr 2025, Shu et al., 2022, Qiu et al., 2024).
• Net Load Synthesis in Power Systems: Incorporation of physics-based PV performance models as conditional modules within the denoising network, ensuring energy-balance and generalization (Zhang et al., 2024).
• Plasma Edge Turbulence, Structural Optimization, Vehicle Dynamics: Unconditional and conditional sampling on high-dimensional output fields, respecting known conservation laws and PDE constraints (Zeng et al., 29 Jan 2026, Bastek et al., 2024).

Evaluation typically focuses on fidelity (MSE, SSIM, LPIPS, edge accuracy), physics-residual minimization, generalization on out-of-distribution conditions, convergence acceleration, and empirical ablation of physics guidance mechanisms.

5. Physics Discovery and Model Analysis

A key conceptual advance in PILD is the capacity for physics discovery:

• Operator Emergence: PILD, trained solely on boundary-to-field pairs, is able to generate field solutions that encode implicit governing operators, with cross-attention and relevance maps aligning closely with known finite-difference/PDE stencils (e.g., Laplacians).
• Layer-wise Relevance Propagation (LRP): Analysis of attention weights and spatial encodings using LRP can reveal which spatial relationships or input sketches most strongly determine output features, suggesting the ability to recover kernel-like structures or conservation laws without explicit equation encoding (Camburn, 13 Jul 2025).
• Cross-domain Generality: The conditional architectures enable extension across multiple physical domains, facilitating knowledge transfer and reuse for classes of PDE-governed systems.

Such operator emergence contrasts with Physics-Informed Neural Networks (PINNs), which require prior hand-specification of the physics to be imposed (Zeng et al., 29 Jan 2026, Camburn, 13 Jul 2025).

6. Comparison to PINNs and Classical Methods

PILD is distinguished from conventional PINNs and finite-difference approaches in several dimensions:

• Equation Agnosticism: PILD does not require manual encoding of the physical operators in the loss; it learns relationships directly from data, enabling generalization and discovery.
• Direct Steady-State Synthesis: PILD bypasses temporal integration and iterative time-stepping, producing steady-state solutions in a single generative pass, which eliminates stepwise error accumulation.
• Sampling and Interactivity: Latent-space or transformer-based generative sampling in PILD enables real-time and interactive exploration, with inference runtime reduced to seconds versus minutes or hours for conventional solvers.
• Generalization and Robustness: PILD models demonstrate robust performance under OOD conditions, domain shifts, and even sparse/broken input regimes due to the fusion of data-driven and physical constraints (Camburn, 13 Jul 2025, Shu et al., 2022, Jia et al., 31 Jan 2025).
• Physics Discovery Potential: Whereas PINNs and classic solvers are restricted to validating or enforcing known physics, PILD’s neural representations can reveal, post hoc, new operator structures and conservation law patterns (Camburn, 13 Jul 2025).

The following table summarizes these differences:

Aspect	PILD	PINN/FDM
Physics Encoding	Discovered, via data	Hand-coded in loss
Temporal Integration	None for steady-state	Required
Generative Output	Direct field sampling	Deterministic solvers
Physics Discovery	Emergent/internal mechanisms	Not present
Multi-domain Universality	Model reuses across physics regimes	Separate models/domains
Sampling Speed	Seconds (latents/transformer)	Real-time not feasible

7. Limitations, Open Challenges, and Future Directions

Despite strong empirical performance, PILD presents open challenges:

• Residual Bias and Jensen Gap: Some bias remains between the score-based marginal and residual-minimized sample (Jensen gap), partially alleviated by advanced DDIM sampling but not fully removed (Zeng et al., 29 Jan 2026).
• Physics Weight Tuning: Hyperparameter trade-off between denoising and physics loss schedules requires problem-dependent tuning.
• Computational Overhead: While inference is efficient, intensive training (hundreds of epochs on GPUs) and per-step residual computation remain nontrivial, especially with complex residual operators or large-scale boundary encoding (Camburn, 13 Jul 2025).
• Extension to Multi-Physics and Control: Formal multi-physics, stochastic systems, and actively controllable generation (e.g., for bifurcation analysis) remain open research avenues.
• Alternative Physics-Guidance Mechanisms: Exploration of heavy-tailed residual distributions, meta-learned penalties, or fully closed-form physics-guided sampling schemes is ongoing.

In conclusion, Physics-Informed Learning via Diffusion constitutes a paradigm-shifting synthesis that leverages the expressivity of score-based generative approaches and the rigor of physical law enforcement or discovery, establishing a universal, efficient, and interpretable class of surrogates and simulators for a rapidly broadening class of scientific and engineering domains (Zeng et al., 29 Jan 2026, Camburn, 13 Jul 2025).